An incident wave creates a discontinuity in the acceleration of the shock front. The amplitudes of the reflected and transmitted waves are also determined. Special attention is given to the case of the weak shocks and the characteristic shocks.

It is shown that there exists an uncountable number of rings R with the properties (1) if r∈R then 2r = 0 and r2 = 0, (2) if r ∈ R then there exist s, t in R with r = st.

Functional and functional differential equations in a Banach space X are related to systems of operators A(t) in C = C(−r, 0; X), given by

Conditions are sought on F such that A(t) generates an evolution system U(t, s)φThis system gives the segments of solution for φ in a certain domain which is determined.

This is a contribution to the general problem of determining the extent to which the geometry of a riemannian manifold is determined by properties of its geodesic spheres. In particular we show that total umbilicity of geodesic spheres determines riemannian manifolds of constant sectional curvature; quasi-umbilicity of geodesic spheres determines Kähler and nearly-Kähler manifolds of constant holomorphic sectional curvature; and the condition that geodesic spheres have only two different principal curvatures, one having multiplicity 3, determines manifolds locally isometric to the quaternionic projective spaces. The use of Jacobi vector fields leads to a unified treatment of these different cases.

Let P(x) and Q(x) be n × n matrices whose entries involve a single given function q(x). Under suitable conditions, estimates

of Liouville-Green type are obtained for the components yν(x) of solutions of the system

In the case n = 2, the estimate is applied to systems arising from fourth-order equations and a generalization of a result of Eastham [2] is obtained.

The theory of singular left-definite canonical eigenvalue problems treated by Nieβen and Schneider in is generalized to arbitrary λ∈(ℂ\ℝ∪{0}. In this enlarged theory the Green's matrix of the problem is evaluated and a natural analogue of the Titchmarsh-Kodaira formula is proved. This formula permits the explicit computation of the spectral matrix playing the main role in the expansion theorems of this theory.

The theories of fractional calculus and of the Hankel transform developed in for the spaces F′p, μ of generalised functions are used to study distributional analogues of dual integral equations of Titchmarsh type. These are shown to have infinitely many solutions in F′p, μ under very general conditions on the parameters involved. These results are used to study the corresponding classical problem in weighted Lp spaces. Existence and uniqueness of classical solutions are investigated and examples given of both uniqueness and non-uniqueness for the classical problem.

Let L be a formally self-adjoint linear differential operator of order m with strictly positive leading coefficient and let m = 2n + 1 if m is odd, m = 2n if m is even. Let y1, y2,…, yn be n given mutually conjugate solutions of Ly = 0 on I, where I is some interval, whose Wronskian is non-zero on I. Then L = (−1)nQ*Q or L = (−1)nQ*DQ where Q is a differential operator of order n, Q* is the adjoint operator and D denotes differentiation. This fact is used to construct further solutions yn+1,−, ym of Ly = 0 so that y1,…, ym is a basis for the solutions of Ly = 0 and for which yi and yn+j are mutually conjugate if i ≠ j. If y1 ≠ 0 on I the degree of L may be lowered by 2 to obtain a formally self-adjoint operator L1 for which mutually conjugate solutions are constructed. If this process is continued a factorization result is obtained which is related to a result of Pólya.

We prove existence and uniqueness results for the solution of nonlinear elliptic boundary value problems, where the linear part of the equation is given by a second-order elliptic operator not in divergence form.

We consider the operator L[y] = y(4) + ((ax2 + bx + c)y′)′ + dy on the half-line [0, ∞). This paper shows that the deficiency indices are independent of the real numbers b, c and d when a ≠ 0. They depend only on the sign of a and are (2,2) if a < 0 and (3, 3) if a > 0. In the case a =0 the sign of b must be considered.

Sufficient conditions are developed for asymptotic stability of the autonomous linear functional differential equation of retarded type. If the asymptotic stability of

implies the asymptotic stability of

then these conditions are also necessary. Necessary and sufficient conditions are developed for the largest cone in the region of stability. These results are illustrated with the example

We consider the formally self-adjoint 2mth-order elliptic differential operator in ℝn given by where lt is an operator of order t, and establish conditions under which the operator on is essentially self-adjoint in L2. A feature is that the major conditions have to be imposed only in an increasing sequence of annular regions surrounding the origin.

Let B be a complete Boolean algebra of projections on a complex Banach space X and let (B) denote the closed algebra of operators generated by B in the norm topology. It is shown that there is a complex Hilbert space H, a complete Boolean algebra B0 of self-adjoint projections on H, and an algebraic isomorphism of B onto B. This isomorphism is bicontinuous when B and B are endowed with the norm topologies, the weak operator topologies or the ultraweak operator topologies. It is also bicontinuous on bounded sets with respect to the strong operator topologies on B and B. As an application, it is shown that the weak and ultraweak operator topologies in fact coincide on B.

A weighted, formally self-adjoint ordinary differential operator l of order 2n is considered, and conditions are given on the coefficients of l which ensure that all self-adjoint operators associated with l have a spectrum which is discrete and bounded below. Both finite and infinite singularities are considered. The results are obtained by the establishment of certain conditions which imply that l is non-oscillatory.

In this work we present an algorithm for computing an integrable function almost everywhere on (0,1) when its moments are known. The method is based on the use of certain delta-shaped sequences, and can be adjusted to take advantage of the local smoothness of the function.

As an application, we give an algorithm for the pointwise inversion of the Laplace transform which utilizes the values of the image function at equidistant points.

It is established that under certain restrictions the solution u of the equation uxy − gu = 0 satisfying u(x, 0) = p(x) and u(0,y) = q(y) in ([0, = ∞) × [0, ∞), changes sign in where (X, Y) is any point in the relevant region.

This paper is devoted to the study of some non linear Schrödinger equations in two dimensions, arising in non linear optics; in particular, it is concerned with solutions to the Cauchy problem. The problem of global existence and regularity of the solutions, the asymptotic behaviour of global solutions, and the blow-up of non global solutions are studied.

A collisionless gas flows through a finite rectangular duct which reflects molecules diffusely. The transmission probability Q of the duct involves the solution of a pair of coupled integral equations. Complementary variational principles have been employed which supply upper and lower bounds to Q. Numerical calculations have been made for a variety of duct shapes and compared, where appropriate, to those of other authors.